You need these things:
o 12V or 24V power-supply
o A socket for a 3.5mm phone-jack, or any other connection from the serial-port to the microcontroller to program the picaxe
o 1 pushbutton and 1 toggle-switch, or 2 pushbuttons
o 1 IC7805 with capacitors, this converts us the 12V or 24V to the 5V we need to operate the microcontroller
o 1 IC ULN2803A, This is a Darlington Transistor Array for the direct use on TTL-Level outputs. Alternatively use 8 single Darlington-Transistors with suitable resistors but it also works with the standard BC547-transistors.
o 1 High-Power FET like the IRF520, or some other Power-Darlington-transistor like the BD649
o A whole bunch of LEDs, different colors like red, yellow, white, warmwhite, blue and ultraviolet. Read step 4 for further informations.
o 1 10k&-potentiometer, preferable with a long knob
o 1 300&- potentiometer for testing purposes
o Some resistors, some cables, a board to build the circuit and of course a soldering-iron
o A measuring tool for currents would also be handy, but is not absolutely necessary
Depending on the power-source you use you might need additional connectors and a housing for the LEDs. I used an acrylic board which I fixed to the housing of the Power-supply.
In older computer-mouses with D-Sub-connectors you might find a good substitute for the phone-jack cable used to program the picaxe.
Picaxes and a lot of other useful stuff might be bought here:http://www.rev-ed.co.uk/picaxe/
For the rest just check out your local dealer.